Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-1,1,0]),K([4,-4,-2,1]),K([-2,0,1,0]),K([26,184,19,-37]),K([-124,113,43,-27])])
gp: E = ellinit([Polrev([-2,-1,1,0]),Polrev([4,-4,-2,1]),Polrev([-2,0,1,0]),Polrev([26,184,19,-37]),Polrev([-124,113,43,-27])], K);
magma: E := EllipticCurve([K![-2,-1,1,0],K![4,-4,-2,1],K![-2,0,1,0],K![26,184,19,-37],K![-124,113,43,-27]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-5a)\) | = | \((-a^3+a^2+4a)\cdot(-a^3+2a^2+3a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1785a^3+6615a^2+5274a-11184)\) | = | \((-a^3+a^2+4a)^{4}\cdot(-a^3+2a^2+3a-4)^{20}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 7724761962890625 \) | = | \(3^{4}\cdot5^{20}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{51150296581786130426003}{286102294921875} a^{3} + \frac{32230691277422593955829}{95367431640625} a^{2} + \frac{24823185420593409932434}{57220458984375} a - \frac{81174148800215168575921}{286102294921875} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(6 a^{3} - \frac{19}{3} a^{2} - 25 a + 15 : \frac{71}{3} a^{3} - 37 a^{2} - \frac{341}{3} a + 78 : 1\right)$ |
| Height | \(0.60129146153295233458978616249096676092\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{5}{2} a^{3} + \frac{13}{4} a^{2} + \frac{49}{4} a - \frac{25}{4} : \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{13}{4} a + \frac{27}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.60129146153295233458978616249096676092 \) | ||
| Period: | \( 10.522334512562749639899133271781489902 \) | ||
| Tamagawa product: | \( 40 \) = \(2\cdot( 2^{2} \cdot 5 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.54239027220219 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((-a^3+2a^2+3a-4)\) | \(5\) | \(20\) | \(I_{20}\) | Split multiplicative | \(-1\) | \(1\) | \(20\) | \(20\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
15.1-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.